Question: Simplify and expand the following expression: $ \dfrac{1}{n + 4}- \dfrac{5}{n - 9}+ \dfrac{1}{n^2 - 5n - 36} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{1}{n^2 - 5n - 36} = \dfrac{1}{(n + 4)(n - 9)}$ Now we have: $ \dfrac{1}{n + 4}- \dfrac{5}{n - 9}+ \dfrac{1}{(n + 4)(n - 9)} $ The least common multiple of the denominators is: $ (n + 4)(n - 9)$ In order to get the first term over $(n + 4)(n - 9)$ , multiply by $\dfrac{n - 9}{n - 9}$ $ \dfrac{1}{n + 4} \times \dfrac{n - 9}{n - 9} = \dfrac{n - 9}{(n + 4)(n - 9)} $ In order to get the second term over $(n + 4)(n - 9)$ , multiply by $\dfrac{n + 4}{n + 4}$ $ \dfrac{5}{n - 9} \times \dfrac{n + 4}{n + 4} = \dfrac{5(n + 4)}{(n + 4)(n - 9)} $ Now we have: $ \dfrac{n - 9}{(n + 4)(n - 9)} - \dfrac{5(n + 4)}{(n + 4)(n - 9)} + \dfrac{1}{(n + 4)(n - 9)} $ $ = \dfrac{ n - 9 - 5(n + 4) + 1} {(n + 4)(n - 9)} $ Expand: $ = \dfrac{n - 9 - 5n - 20 + 1}{n^2 - 5n - 36} $ $ = \dfrac{-4n - 28}{n^2 - 5n - 36}$